Bandages, I had a some big bandages, and an IV, and my room didn't look like any jail or prison.
Sterile walls with nothing on them. An actual door with door handle.
"Relax," a man in a black suit coat and red tie said. He was bald, with a clipboard in his hand. "Your life isn't over. As a precaution, we have temporarily instituted the Case Delta 9, which prevents any Dungeon doors from opening for you.
"Any math you try more advanced that the Novitiate level, will elude you somehow. I am James Mckinnon of the National Security Talent and Threat Monitoring Center. I evaluate prodigies to see if their skills may be able to benefit this country, or if they are a threat to the nation.
"Any more stunts like that Earthquake will land you on the mandatory kill list, and I won't have any power to chance that. Some would like to see you made a political monkey already, but we desperately need help to pass Journeyman O-3.
"As you've seen already, the nature of the Dungeon's ranking system changes over time. Novitiates get the cool and kid friendly belts system. We didn't design the system, we just trained kids to associate black belts with cool."
"Initiates are sea life, as you've seen, and algebra and geometry enter the picture. Despite my initial peg on you, our sensors detected a level 6 geometric anomaly with an associated unknown maths sensor reading. Your weapon to take out a monster was somehow unknown and geometric."
"Now before I grill you on this mathematical weaponry, I want to give you the carrot. If you pass through this process to my satisfaction and my superior's satisfaction, you will receive room and board in the NSTTMC Talent Housing System. You will have access to continue your pursuit through the dungeons, but with a disciplined, threat and response understanding of appropriate firepower.
"You will receive pay, and be able to acquire some civillian goods, but 50% of your pay will be garnished to pay for the damage you have done to both Novitiate and Initiate dungeons, until that full amount is paid," the man said. "How do you feel about that?"
I sighed in relief, couldn't really think about the long term, my brain wasn't thinking in an advanced mode. "Relieved that I can still take on the dungeons. I'd like to learn from your best too."
James said, "Until you reach Journeyman E-1, you won't be able to associate with the other members of our Talent team. Forgive me, but that is the rule that President Phearson put done. Your progress through the rest of Initiate dungeons and Apprentice dungeons will allow us to evaluate your value as a national asset.
"But lets back up. I'm sure you know the alternatives here. Please, give us an account of what lead to the Geometric Unknown anomaly."
I described my progress through the system, being a Dolphin stage initiate, taking on the Trolls and Kobolds, getting the perfect score, writing down the sides of a regular n-gon, and then the fateful hyper-volume question.
"I'm going to reduce the drip on this, so I can have your full faculties to answer in detail," said Agent James.
After another half hour, in which the Agent went to get coffee and confer with superiors, he came back in. "If you did try to attack with your unknown mixture, this building more take damage, but we are prepare, and you would lose your chance to take on the dungeon."
"I won't say any mathematics without permission, I swear," I said, just wanting to get back to my dungeons in peace.
"What would you call the mathematics you used in the Anomalous question?" James asked.
"Multi-dimensional geometry (beyond 3 dimensions). Arbitrary dimensional geometry. Multivariate Integral Calculus. Multiple Integrals."
"If you reach Journeyman E-1, we'll have you show your work in a damped sealed chamber."
I pondered why they considered some multiple integrals to be so dangerous. It's just the proper choice of bounds of integration across an indefinite number of integral signs. And the Gamma function once you see the generalization.
"Senors just read a buildup of Unknown, please stand down.," James shouted.
."Sorry," I said. "Can you give me a summary of safety practices with your department. What kind of things are dangerous? Is even the thoughts in my mind dangerous?"
"Our advanced personalized sensor suite detects mathematical thoughts before they can be spoken or written. We used this with those we classify as a danger, or those who are still unknown, like yourself. We have many wards full of mathematicians, who are, as you say, break glass in case of emergency. If they start thinking mathematical thoughts, we administer sedatives.
"Unfortunately, a mathematical mind deprived of opportunity to think about math becomes atrophed after a few months. Fortunately, this renders these dangerous individuals safe after a six month stay in the retention wing of NSTTMC.
"Just thinking mathematics is not dangerous, but please do not think about Unknown level mathematics without prior approval. It makes us think you are a terrorist.
"In practice, as an operative, you will operate on a 4 stage model. Stage 1 is general purpose mathematics. You may use these in planning sessions and to keep track of metrics on mission success.
"Stage 2 is general combat mathematics. These are allowed for use when in a dungeon in combat. Stage 3 is pre-authorized mathematics. Each type of stage 3 mathematics is covered by a specific code word. If our planning department determines it is needed, they will brief you for usage of the restricted mathematics for a specific mission component.
"You must be qualified for use of all Stage 3 mathematics. Since these mathematics can't be taught without grave risks, you must already know them to be cleared.
"Stage 4 is restricted mathematics. You may never use Stage 4 mathematics. All unknown and unclassified mathematics is considered Stage 4 until reclassification."
I didn't consider myself a mathematician. I was a recreational mathematician on Earth, with memories of college courses, especially calculus and numerical analysis.
I let my mind clear of math, when I saw James tense up again.
"Did you have other questions, or information I need to go through?" I asked. "I'm willing to cooperate. I don't want to lose mathematics, even if I can't use it freely."
"Our neural brain sensor needs to be calibrated, it seems. Your thought patterns can raise to a level 3 warning in an instant. This neural sensor is also how we evaluate our people's capability and additional stage 3 codes. Each person has their own stage 3 codes, which we subdivide based on the sensors.
"I will ask you to start slowly, with basic arithmetic, and then gradually think of new mathematical knowledge you contain, until you can't think of anything else."
James nodded to me.
I thought of basic arithmetic, not the structure which brought its similarity, but just addition problems with single digits.
Subtraction, then arithmetic with 0 and 1. Multiplication. Division.
"Stage 2" James said.
I thought about algebraic properties of linear functions.
"Too much, Edmond. Stage 2."
I solved x + 2 = 4. 2 x = 7. I observed angle and parallel lines. Scaling and reflecting. 3 digit arithmetic.
"Stage 3. Classification #1"
I thought about linear functions again, and their generalization in linear algebra. Vectors. Eigenvalues.
"There's a least 3 codes. Take 1 thing at a time."
So I tediously directed my thoughts until I was thinking on only one little piece of linear algebra at a time.
The same proceeded with calculus, and trigonometry, seperating them into dozens upon dozens of little code worded pieces.
"Stage 4. Thoughts only!"
I thought on the type of integrals involved to calculate the hypervolume of n dimensional unit balls again.
Circumference of a radius n circle is 2 pi n. To get the area of the ball, Integrate circumference(n) dn from n = 0 to r.
Int_0_r 2 pi n dn = [pi n^2] : 0_r = pi r^2
I racked my brain thinking of how to derive the nth dimensional unit ball volume in the general case.
You could integrate it radiating out from the center, or across a hyperplane slice, but I was struggling with exactly the details.
Integral of the Sqrt(1-x^2) had an arcsin in it. Polar coordinates, how to generalize to more than 3 dimensions. There were lots of u substitutions, and working out anti-derivatives.
"Stage 5," James said.
"You didn't mention Stage 5 before," I said.
"Stage 5, the highest encountered Stage of mathmetics amongst the Dungeon denizens. No one has reached Stage 5 on our sensors. Stage 5."
I went for mathematics that I barely understood on an only recreational level.
There's an isomorphism between the hands of a clock and the group of numbers under addition mod 12. An isomorphism implies we can do clock math on the clock face, then make the translation to the mod group, or translate to the mod group, do the arithmetic and translate back.
You can establish an isomorphism between a mod 86400 and a stacked set of clock hands, or a set of tied together hours mod, minutes mod and seconds mod groups.
Or you can devolve your mod 60 group into two mod 2 groups, a mod 3 group and a mod 5 group. So the structure of groups available matches the structure of primes.
On a certain level, the groups themselves form an isomorphism of the natural numbers, with the operator of multiplication.
"That's stage 4," said James. "Try for Stage 5."
Flip it on its head. What do I know especially well?
We can sometimes have a functional analogy to squares and square roots. A functional square is the composition of a function from R -> R (firstly, but also C -> C, R^n -> R^n), with itself
Fsqr of e^x is e^e^x.
But what is the Fsqrt? It would be the function f(x) such that f o f (x) = e^x
No standard function fits.
The Fsqrt of any linear function is another linear function.
if g(x) = a x + b
f(x) = c x + d
f o f (x) = c (c x + d) + d = c^2 x + cd + d
c^2 = a
cd + d = b
c = sqrt(a)
(sqrt(a) + 1) d = b
d = b / (sqrt(a) + 1)
g(x) = sqrt(a) x + b / (sqrt(a) + 1)
Any monomial has another monomial as its functional sqrt, in a similar fashion.
f(x) = d x ^ c
g(x) = b x^a
d (d x^c) ^ c = d d^c x^(c^2)
d d^c = b
c^2 = a
c = sqrt(a)
d d ^ sqrt(a) = b
log(d) + sqrt(a) log(d) = log(b)
(sqrt(a) + 1) log(d) = log(b)
log(d) = log(b) / (sqrt(a) + 1))
d = e^(log(b) / (sqrt(a) + 1))
g(x) = b^(1/ (sqrt(a) + 1)) ^ sqrt(a)
or
g(x) = b ^ (sqrt(a) / (sqrt(a) + 1))
x is its own functional square root, but so is -x, and 1/x, and any other function which is its own inverse.
More generally, we can look at Taylor series.
Any function where f(0) = 0 is easier to work with, because the constant term makes the expansions tough if it is non-zero.
Formal series actually, because we don't care too much about convergence properties yet.
if f(x) = a0+a1 x + a2 x^2 + ...
if a0 is not 0, we can say b0 = 0, because b0 doesn't contribute to any later terms then, and b0 = 0, so the constant term of the iterated expression is 0.
g(x) = b0+b1 x + b2 x^2 + ...
g(g(x)) = b0 + b1 (b0 + b1 x + b2 x^2 ...) + b2 (b0 + b1 x+ b2 x^2 + ...) ^2 + ...
if b0 is non-zero, we can get all powers from each subterm, by taking some number of x^0 parts.
If b0 is zero, then the linear subterm must only have coefficients of 1 or higher, the quadratic subterm is quadratic or higher and so on.
b0 = a0 = 0
g(g(x) [partial] = b1 (b1 x + b2 x^2 + ...) + b2 (b1 x + b2 x^2 + ...) ^ 2 + ...
g(g(x) [linear] = b1^2 x
g(g(x) [quad] = b1 b2 x^2 + b2 b1^2 x^2
and so on.
Only finitely many terms can contribute, so in theory we can gradually and eventually solve for every coefficient of g(x).
Now if f(x) = e^x - 1, a0 = 0. a1 = 1, a2 = 1/2!, a3 = 1/3!, etc. The b0 calculations would then give us successive approximations of (e^x -1) fsqrt.
James finally spoke up, "Ok that's enough. Our Monster outbreak scale sensor spiked at 7 kilo-amperes."
A true professional mathematician could blow these guys minds away. I was primarily a programmer. Curry-Howard Isomorphism meant from a certain point of view, you could call what I did mathematical, but practically it wasn't. I was familiar with tools that used the Homotopy Type Theory, familiar in the sense of being just barely past a novice in them.
"Localized spike. 73 kilo-amperes. Give your brain a rest, Edmond."
I relaxed, leaning back in the chair I was strapped to.
"I'll be back. There's some negotiations to be had."
<> o <> o <>
Name: Edmond Bauer
Ranking Status: Initiate (Stingray)
Sphere Ranking: 66,855,273rd (of ~121,000,000) (45%)
Lives: 29
Race: Elothean
Race: Streak 3 - Skip three enemies with full rewards for every eleven you defeat without getting defeated.
Class: N/A
Class Bonuses: None UnlockedTracking:
Algebra (stingray)
Geometry (Lanternshark)
<> o <> o <> o
Dungeon doors were shifting to provide access to the Initiate geometry doors.
"Edmond Bauer has the potential unlock new realms of safe scientific inquiry and technical advancements. The dungeon has always provided us with new wonders when we have plumbed its depths before. When Charles Hillary was the first to reach Journeyman E-1, our civilization was brought into new light, as we learned the principles of electricity When our team reached O-1, modern life was possible. O-5 and beyond beckons, gentlemen," Jim said to the other gathered members of the NSTTMC.
"The Case Analysis is our best bet. If Edmond can restrain himself an achieve a 90 or above, I'm willing to overlook his past discrepancies," said Jim's supervisor.
"Past potentials have rarely scored that high. It’s difficult, if you have the capacity, to go through that.”
“Let’s brief him, and then start.”
< program start: Algebra stingray>
~
The start of the algebra plan was 5 eels making him do substitution of variables with fractional values.
Without paper it took a little thinking but it wasn’t bad. He was only allowed tier 2 attacks in algebra while doing these algebra sections.
Next there was mixed set of problems, some evaluations, laughably easy square roots, and recurrence relations that seems way too advantageous until I realized they were just linear equations presented in a funny way.
I got a chalkboard for these next problems:
4a = -3 + 3.25 a
0.75 a = -3
3a = -12
A=-4
2,
4
BBB
M
BBB
M
-14=7/2 n
n=-4
-6=0.5 d
D=-12
5
-3 2/3 p = -11
11 p = 33
P=3
16,
2
2
JIM: “Remember your restrictions in the challenge dungeon.”
Little baby sharks charged at me as o wrote on the chalkboard.
“23x = 3x + 60 -> x = 3
Minor impact.
x^2-a^2 = (x-a) (x + a)
Bigger
Let’s not go too big.
X^2+13x-30 = 0 : (x+15)(x-2)= 0
I stuck to quadratics and big lineare and advanced to electric eel immediately.
